A footrace: slope and systems of equations

I taught a lesson about slope and systems of linear equations after the style of Dan Meyer’s “What Can You Do With This” series. I videotaped a footrace between myself and some of my students, and showed it in class.

I tried to follow several of Dan’s ideas about a good video segment like this. There is no clear focal point specified by the video, and no imposed frame of reference. I decided not to show the beginning or end of the race in the initial video with the hopes that the omission would incite interesting questions.

The startup question was simply, “who wins?” Some students immediately identified the runners that hit the right edge of the screen as the winners, and a few asked where the finish line was. My plan at this point was to let some controversy mount and then steer the conversation a little, but to my delight, in the midst of the conversation, a student in the back suggested we just figure out who is going faster. So, our question was narrowed. I took the opportunity to point out that this is what problem solving is about – figuring out what questions to ask. I somewhat artificially changed the student’s question to “how fast are the different runners going?” and challenged the class to figure it out.

Which is actually a pretty complicated task, given only the video above. I was using the quicktime player to show the video, which has a pretty great slider control that can practically move between individual frames. But when some students suggested that we need a timer, and another group said that we’d also need a frame of reference, I pulled up a new version.

I also had a video with just a timer and another with just the line ready in case they only asked for one of them.

From here, the conversation led itself at what felt like a fantastic pace. There was some controversy about whether or not we could really find a speed by just finding the slope between two points, and about what points to use, and whether or not it mattered. We checked with two different kinds of calculations, and got similar speeds, so the kids were convinced (though this sort of proof makes me feel a little dirty).

Somebody said, “OK, so who wins here?” and another student responded, “It depends on where the finish line was.” Again, I pointed out the great problem solving going on here (I like to emphasize that coming up with a question is just as important as being able to answer it). You can guess where this is going – we found the point at which it would be a tie using a system of linear equations.

I gave the video to the kids to play with themselves on individual computers. For an honors assignment I asked them to figure out how much of a head start I gave the other pack of racers, in feet, and several of them chose to tackle the problem. I considered the class a success because the students liked it, and I had everyone practicing solving systems of equations with interest and a purpose. And for those students that steered the conversation with narrowing questions (and many more with viable alternate questions) it was good practice problem solving. The lesson took 30 minutes.

There are several ways to improve this lesson. I didn’t have video of the finish or the start, to give the class closure, the proof that they were right. That’s not always bad, but it would have rounded this lesson off better, and I think it’s important to have kids leaving with as much satisfaction as possible. Also, some of the students in the race were also in my class. They knew the outcome of the race beforehand (to a finish line not specified in the video), and they knew that I had given them a head start, etc. I can’t wait to use this video next year (although I made the mistake of including a freshman who will probably be in my class eventually). This definitely affected our conversation in class a couple of times. Then again, maybe it’s worth it to have actual class members running in the video on screen.

This lesson inspired by Dan Meyer, Mr. Sweeney, and a newfound category of interesting situations for class. Please leave comments, criticisms, comparisons, and suggestions for improvement!

6 thoughts on “A footrace: slope and systems of equations”

This sounds like a good lesson. I’m wondering, though, where the idea of using slope between two points to represent speed came from. It sounds like a very teacher-y thing. As a student, I would look at your number line and tell you that the slope between two points is zero, because your line is horizontal. (Actually, if I was one of my students, I would probably say it was undefined, or 7, or -5)

So where does the idea that one coordinate is time while the other is distance come from? If it’s a student construction, I would love to hear how you laid the groundwork for them to be able to do that. If not, then I wonder, is it important for them to come up with that idea? Do they just naturally reach for graph paper when confronted with a dynamic problem like this?

I really liked that you didn’t include the starting line or the finish line or the point where you pass anyone. I think it’s the realization that something interesting is about to happen just off camera that creates the interest in your students.

Thanks for the comments, Jason. You’ve made me start analyzing whether “speed as slope” is an important concept! This is an Algebra 2 class, so on one hand I’m thinking it’s good to stretch their understanding of the concept of slope. On the other hand, this is me being a math dork – I think it is SO COOL that slope can mean different things, and that you can read it right off of graphs to see where something is going fast, slow, etc.

This lesson was not our class’ first contact with this concept. Early in the year I talked about graphs with different dimensions for axes, with a big focus on space-time graphs. Instead of introducing the slope directly, we drew space time graphs with 4 or 5 points on it, like (4s, 10m) and (6s, 20m). Then, on a sidewalk I had prepped with 5m increments marked in chalk, the students had to get to those points (in a group of 4, which was pretty hilarious to watch). Coming back to the class, we talked about the experience, and some kids mentioned speed between certain points. None of them came up with the language of “slope,” but when I suggested it most seemed to latch on to it. Some didn’t. However, by the end, everyone could point to a steep line and identify it as faster than a shallow line. Some just couldn’t say “that one has a higher slope.”

Awesome lesson! Love it. And, for the record, speed as slope is very important. First of all, from a purely pragmatic standpoint… well, I teach calculus, where an understanding of the connection between speed and slope is necessary for at least a third of what goes on. But more fundamentally, I think you’re right that it can only enrich their understanding of slope to see it as an expression of a speed. And also, if it gets you excited, why put a damper on that? It, and you being excited about it, might get them excited too…

Nothing to add here, Riley, except my congratulations on doing justice to wonderful, messy problem solving. You let your students, more or less, run the show.

And, not that it needs saying, but how cheap are these kind of digital learning experiences? You can bring the world into your classroom and tweak it in useful ways — adding axes and a timer — all at the cost of, I dunno, your time? Zero dollars? It’s great.

Thanks for the compliments, guys. Strangers on the internet commenting on my work gives me a bigger rush than I expected!

@Ben: I tend to agree that if it gets me excited I should run with it. Many of my students remark favorably about my enthusiasm on evaluations of my class. However, I know that I have a propensity towards running with something too far when I like it, and if I’m not careful I can forget to focus on the students. Now, in the case of slope, I happen to think that it is vital for students to understand that it can represent things above and beyond “steepness,” so I don’t really plan on cutting it out. But when I want to derive the chain rule in front of the whole class because I think it’s so neat… it takes a little more thought.

Software: Adobe Aftereffects (I use open-source software for almost everything, but could not find an alternative (even commercial) to AE for this kind of project). I was lucky that my school already had a license for this, and that I could install it on my laptop. This was my first time using the software (or any video-editing software) and I was surprised by how easy it was to learn. Including learning and production, the videos above took me about 1 hour. I recently did a similar timer overlay on another video in about 5 minutes, and I expect times to continue decreasing.

So, from the lucky position of having hundreds of dollars equipment already available to me, this project was zero extra cost and maybe 2 hours of work.